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Online Social Networks and Media Navigation in a small world

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Small world phenomena Small worlds: networks with short paths Obedience to authority (1963) Small world experiment (1967) Stanley Milgram (1933-1984): “The man who shocked the world”

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Small world experiment Letters were handed out to people in Nebraska to be sent to a target in Boston People were instructed to pass on the letters to someone they knew on first-name basis The letters that reached the destination followed paths of length around 6 Six degrees of separation: (play of John Guare)

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Milgram’s experiment revisited What did Milgram’s experiment show? – (a) There are short paths in large networks that connect individuals – (b) People are able to find these short paths using a simple, greedy, decentralized algorithm

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Small worlds We can construct graphs with short paths – E.g., the Watts-Strogatz model

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Small worlds Same idea to different graphs

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Navigation in a small world Kleinberg: Many random graphs contain short paths, but how can we find them in a decentralized way? In Milgram’s experiment every recipient acted without knowledge of the global structure of the social graph, using only – information about geography – their own social connections

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Kleinberg’s navigation model Assume a graph similar (but not the same!) to that of Watts-Strogatz – There is some underlying “geography”: ring, grid, hierarchy Defines the local contacts of a node Enables to navigate towards a node – There are also shortcuts added between nodes The long-range contacts of a node Similar to WS model – creates short paths

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Kleinberg’s navigational model Given a source node s, and a navigation target t we want to reach, we assume – No centralized coordination Each node makes decisions on their own – Each node knows the “geography” of the graph They can always move closer to the target node – Nodes make decisions based only on their own contacts (local and long-range) They do not have access to other nodes’ contacts – Greedy (myopic) decisions Always move to the node that is closest to the target.

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Example

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Long-range contacts

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Clustering exponent This exponent is the only one for which we can obtain “short” (polylogarithmic length) paths

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Theoretical results

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Proof Intuition The algorithm has the same probability to link to any scale of resolution – logn scales, logn steps in expectation to change scale

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Proof intuition The algorithm is able to replicate what happens in the Milgram experiment

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Long range links in the real world

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Linking by rank

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Live Journal measurements Replicated for other networks as well (FB) Is there a mechanism that drives this behavior?

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Other models Lattice captures geographic distance. How do we capture social distance (e.g. occupation)? Hierarchical organization of groups – distance h(i,j) = height of Least Common Ancestor

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Other models

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Theorem: For α =1 there is a polylogarithimic search algorithm. For α ≠1 there is no decentralized algorithm with poly-log time – note that α =1 and the exponential dependency results in uniform probability of linking to the subtrees

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Generalization

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Doubling dimension

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Small worlds with nodes of different status

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Application: P2P search -- Symphony Map the nodes and keys to the ring Link every node with its successor and predecessor Add k random links with probability proportional to 1/(dlogn), where d is the distance on the ring Lookup time O(log 2 n) If k = logn lookup time O(logn) Easy to insert and remove nodes (perform periodical refreshes for the links)

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Proof of Kleinberg’s theorem

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Normalization constant

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